NOAA Technical Memorandum NMFS-AFSC-169


Probability Sampling and Estimation 

of the Oil Remaining in 2001 from the 

Exxon Valdez Oil Spill in Prince William 

Sound, Alaska 

by 

J. Pella and J. Maselko 

U.S. DEPARTMENT OF COMMERCE 

National Oceanic and Atmospheric Administration


National Marine Fisheries Service


Alaska Fisheries Science Center


April 2007 
NOAA Technical Memorandum NMFS 

The National Marine Fisheries Service's Alaska Fisheries Science Center 

uses the NOAA Technical Memorandum series to issue informal scientific and 

technical publications when complete formal review and editorial processing 

are not appropriate or feasible. Documents within this series reflect sound 

professional work and may be referenced in the formal scientific and technical 

literature. 

The NMFS-AFSC Technical Memorandum series of the Alaska Fisheries 

Science Center continues the NMFS-F/NWC series established in 1970 by the 

Northwest Fisheries Center. The NMFS-NWFSC series is currently used by 

the Northwest Fisheries Science Center. 

This document should be cited as follows: 

Pella, J., and J. Maselko. 2007. Probability sampling and estimation of 

the oil remaining in 2001 from the Exxon Valdez oil spill in Prince 

William Sound, Alaska. U. S. Dep. Commer., NOAA Tech. Memo. 

NMFS-AFSC-169, 60 p. 

Reference in this document to trade names does not imply endorsement by 

the National Marine Fisheries Service, NOAA. 
NOAA Technical Memorandum NMFS-AFSC-169


Probability Sampling and Estimation


of the Oil Remaining in 2001 from the


Exxon Valdez Oil Spill in Prince William


Sound, Alaska


by 

J. Pella and J. Maselko 

Alaska Fisheries Science Center


Auke Bay Laboratories


11305 Glacier Highway


 Juneau, AK 99810-8626


www.afsc.noaa.gov 

U.S. DEPARTMENT OF COMMERCE 

Carlos M. Gutierrez, Secretary 

National Oceanic and Atmospheric Administration 

Vice Admiral Conrad C. Lautenbacher, Jr., U.S. Navy (ret.), Under Secretary and Administrator 

National Marine Fisheries Service 

William T. Hogarth, Assistant Administrator for Fisheries 

April 2007 
This document is available to the public through: 

National Technical Information Service 

U.S. Department of Commerce 

5285 Port Royal Road 

Springfield, VA 22161 

www.ntis.gov 
Notice to Users of this Document 

This document is being made available in .PDF format for the convenience of users; however, the 

accuracy and correctness of the document can only be certified as was presented in the original hard 

copy format. 

iii

 ABSTRACT 

Disputes concerning the amount of spill oil remaining in Prince William Sound, Alaska, 

from the Exxon Valdez grounding in 1989 required revisiting the region in 2001. In contrast to 

earlier surveys which were based on purposeful selection of sampling locations, probability 

sampling was applied in order that unbiased estimates with measures of their precision could be 

computed. Beach segments and subsegments were stratified by their oiling histories and lengths, 

and random samples were selected for a visit. At each beach visited, the surface was grided into 

tidal elevation intervals and perpendicular columns, and every intersecting block was further 

subdivided into quadrats (0.25 m2 in area) for random and adaptive sampling. Adaptive sampling 

pursued oil found in initial random quadrats in order to delimit entire patches. Subsurface oiled 

sediments were classified to a visual scale, and oil present in selected quadrats was extracted and 

weighed in a calibration study. The surface and subsurface oiled areas of the sediments at the 

visited beaches were estimated, together with the weight of oil in their subsurface sediments from 

the calibrated visual scale. Conservative estimates of oiled areas and weights for the visited 

beaches included only oil seen at random and adaptive quadrats, but unbiased estimates were 

computed by expansion for quadrats not sampled. The estimates at visited beaches were 

expanded for unsampled beach segments of strata, and summed for the total in Prince William 

Sound. Precision was evaluated by analytical formulas as well as by bootstrap resampling. 

Unbiased estimation of oiled areas and weights from oil found at the random and adaptive 

quadrats with precision evaluated by bootstrap resampling determined that until 2001, Prince 
iv 

William Sound still had a total of 41,000 m2 surface oiled area (95% interval, 20,700 – 70,500 m2), 

and a total of 71,000 m2 subsurface oiled area (95% interval, 37,700 – 113,200 m2), having 

subsurface oil weighing 50,000 kilograms (50 metric tons (t)) (95% interval, 24.4 - 82.6 t). 

Unbiased estimates based on the random quadrats only were 78,000 m2 for subsurface oiled area 

(95% interval, 40,600 – 127,300 m2), and 56 t (95% interval, 26.1 - 94.4 t) for subsurface oil mass, 

agreeing well with the estimates based on combined random and adaptive quadrats when 

considered in light of sampling error. 
v 

CONTENTS 

ABSTRACT ................................................................. iii

INTRODUCTION ............................................................. 1

METHODS .................................................................. 3

Sampling Design ........................................................ 3

Segments and Subsegements ......................................... 3

Within Subsegment Sampling ........................................ 5

Gravimetric Samples ............................................... 7

Estimation Formulas ..................................................... 8

Oiled-area for a Selected Beach Subsegment ............................ 8

Oiled-area Estimation for Strata ..................................... 10

Oil-weight Estimation ............................................. 12

< 100 m Beach Subsegments ........................................ 14

100 m Beach Subsegments ......................................... 16

Total Weight of Spill Oil Remaining in Prince William Sound ............. 21

Bootstrapping for Determination of Precision of Estimates ...................... 23

Oiled Area, < 100 m Subsegments ................................... 23

Oiled Area, 100 m Subsegments ..................................... 24

Visual-category Composition of Oiled Areas, 

< 100 m or 100 m Subsegments ................................... 25

Average Weight of Oil per Quadrat of the Visual Categories ............... 26

Oil Weight for Strata with Subsegments <100 m ........................ 26

Oil Weight for Strata with Subsegments of 100 m ....................... 27

RESULTS AND DISCUSSION ................................................. 29

CITATIONS ................................................................. 33

TABLES .................................................................... 35

FIGURES ................................................................... 44

APPENDICES ............................................................... 53

INTRODUCTION 

On 24 March 1989, the oil tanker Exxon Valdez ran aground on Bligh Reef in Prince 

William Sound, Alaska, releasing 41 million liters (36,490 metric tons (t)) of Alaska North Slope crude 

oil into the sea (Fig. 1). Subsequent linear estimates for the oil’s dispersal on beaches declined from 

783 km in 1989 to 10 km in 1992 (Neff et al. 1996), while the remaining oil volume in the fall of 

1991 was estimated at 40,882 liters (Koons and Jahns 1992). The traditional methodology to 

estimate volume of oil stranded on shorelines uses visual surveys together with average oil 

depths. The accuracy of both linear and volume estimates depends on the validity of an 

assumption that absence of surface oil on beaches indicates the same condition for subsurface oil 

(Finkelstein and Gundlach 1981, Owens 1987). This supposition is plausible shortly following 

the stranding of oil on the beaches, but as time passes after the spill, surface oil becomes a poor 

indicator of subsurface oil. 

Neff et al. (1996) reported that in 1991, about 2 years after the spill, only 33% of pits with 

surface oil also had subsurface oil, but importantly, no information was available regarding the 

occurrence of subsurface oil without surface oil presence. When residents of Prince William 

Sound complained of oil persistence at a local beach used for subsistence harvest of shellfish, a 

shoreline cleanup in 1996 and 1997 about 8 years after the spill unearthed substantial deposits of 

subsurface oil that were not evident from surface observations (Brodersen et al. 1999). During 

the present survey in 2001, about 12 years after the spill, only 24 of 225 (11%) pits with surface 

oil also had subsurface oil, and more importantly, only 24 of 341 (7%) pits with subsurface oil 

also had surface oil. 

Previous studies targeting subsurface oil have used purposeful selection of sampling 

locations (Gibeaut and Piper, 1998; Neff et al. 1996) in place of long-standing methods of 
2

probability sampling (e.g., see Cochran 1963). Both Gibeaut and Piper (1998) and Neff et al. 

(1996) selected shoreline sampling locations in Prince William Sound based on their oiling 

histories and recommendations from the public. In contrast to probability sampling, purposeful 

selection does not guarantee unbiased estimates, nor does it allow an evaluation of accuracy and 

precision of estimates. Owens (1987), Neff et al. (1996), and Gibeaut et al. (1998) concede that 

sampling crew judgment has a large effect on such survey estimates, especially when applied to 

subsurface oil. Both Neff et al. (1996) and Gibeaut and Piper (1998) used a systematic sampling 

approach at the selected sites to delineate the extent of subsurface oil patches, similar to the 

adaptive sampling plan followed in the present Auke Bay Laboratory survey at randomly selected 

sites. Critical to probability-based estimation with adaptive sampling is the random selection of 

initial pits at a selected site, but both Neff et al. (1996) and Gibeaut and Piper (1998) failed to 

describe how locations of the initial subsurface pits at a site were chosen. 

The use of purposeful selection of sampling units by the different surveys resulted in 

conflicting estimates of oil persistence. Whereas Neff et al. (1996) reported 12,000 m2 of 

subsurface oil remaining in 1992, Gibeaut and Piper (1998) concluded that in 1993, the 

remaining subsurface oil area in Prince William Sound was closer to 33,749 m2. The 1993 survey 

also provided a volume estimate for subsurface oiled sediments adjacent to only the surveyed 

shoreline, equal to 2,041 m3. Neff et al. (1996) estimated oil that persisted on 96 km of shoreline 

in 1991 and 10 km in 1992. Because probability sampling was not used previously by either 

Exxon or public sponsored (Exxon Valdez Oil Spill Trustee Council) scientists, of course, their 

accuracy or precision could not be appraised. The actual amounts of oil remaining was unknown. 

Therefore, the disagreements among scientists over the remaining amounts of oil required 
3

another survey for oil in Prince William Sound, this time using probability sampling so that 

accuracy and precision of the estimates could be evaluated. The Exxon Valdez Trustees Oil Spill 

Council contracted with the ABL in October 2000 to perform the survey and estimation. 

Here, we describe the ABL’s survey sampling design, provide related estimation 

formulas, and report the point estimates and their precision which were computed for the 

dispersal area and quantity of the remaining oil. The dispersal area was determined from a 

stratified random sampling design of beaches in Prince William Sound, based on their oiling 

histories. The oiled surface and subsurface areas of selected beaches were estimated by the initial 

random sampling of smaller comprising units called quadrats, followed by delineation of 

discovered oil patches by systematic search. The quantity, or weight, of oil remaining was 

determined from visual observations of oiling at sampled quadrats in the survey, which were 

transformed to oil weights from data of a calibration study. Two approaches to evaluate the 

precision of estimation are compared—analytical formulas derived from sampling theory 

(Cochran 1963, Thompson 1992), and the computer-intensive method called bootstrap 

resampling (Efron 1982). 

METHODS 

Sampling Design 

Segments and Subsegments 

Since the 1989 Exxon Valdez oil spill, Prince William Sound beaches have been surveyed 

repeatedly for the prevalence of oil. The main surveys occurred in 1989, 1990, 1991, and 1993, 

with the goal to map the geographical extent of oil on beaches and to classify the levels of 
4

contamination. The surveys measured subsurface contamination, or oiling, levels by a coarse 

ordered scale, whose decreasing values were termed “heavy”, “medium”, “light”, “oil film/trace” 

and “clean”. The surveys related uninterrupted horizontal distances of such contamination levels 

to a list of adjacent beach segments. Three separate oiling categories of these beach segments and 

their total shoreline lengths were identified for which our expectations of finding oil were 

highest: 

1) Heavy oiling found during the 1990, 1991, and 1993 surveys, 24.4 km. 

2) Medium oiling found during the 1990, 1991, and 1993 surveys, 49.1 km. 

3) Heavy oiling found during the 1989 survey, but becoming less than heavy oiling more 

recently, 43.1 km. 

Other shorelines of Prince William Sound that had medium and low oiling in 1989 and low 

oiling more recently were assumed to be clean without further measurement in 2001, and 

therefore were excluded from the estimation for total oil persisting in Prince William Sound. 

Since the oiled shorelines were not contiguous, the total shoreline within any oiling 

category consisted of beach segments of varying lengths. These variable length segments were 

then divided into subsegments of 100 m in length or less. This division was necessary because 

the field crews could only sample a beach of about 100 m length between high tides each day. 

Since the segment lengths were not even multiples of 100, leftover subsegments less than 100 m 

resulted. Therefore, each oiling category contained two subcategories: 100 m and < 100 m beach 

subsegments. Overall, six strata of beach subsegments resulted, being the possible combinations 

of two subsegment length categories nested within each of the three oiling categories (Fig. 2). 
5

Beach subsegments to be sampled were selected at random from the substrata. For 100 m 

subsegments, simple random selection without replacement was used. For < 100 m subsegments, 

random selection with probability proportional to length (ppl) was used. The ppl sampling 

requires replacement. 

Within Subsegment Sampling 

Because the oil was not deposited on the shorelines uniformly with respect to tidal 

elevation, the beach surface of each selected subsegment was divided into six 0.5 m tidal 

elevation intervals (0-0.5, 0.5-1.0, 1.0-1.5, 1.5-2.0, 2.0-2.5, 2.5-3.0 m, referring to elevation 

below mean high tide). These intervals covered the beach between the highest elevations that 

deposition could have occurred (+4.8 m above mean low tide) and the lowest elevation 

accessible diurnally to surface sampling (+1.8 m above mean low tide). 

Stratified adaptive sampling (Thompson 1992) was used within subsegments because the 

oil was expected to be distributed in rare patches and mapping these patches would improve 

precision. Such delineation of the oil distribution was consistent with the earlier traditional 

survey methods, but at a much finer geographic scale. In order to distribute the sampling over the 

beach surface of a subsegment, the surface was divided into vertical, equal-width columns. In the 

case of 100 m subsegments, eight columns, 12.5 m wide, were formed. Shorter beach 

subsegments had fewer columns of about the same width. The intersections of horizontal 

elevation intervals and vertical columns on the beach surface resulted in blocks (e.g., 48 blocks 

per 100 m subsegment), each of which was subsampled for the presence of oil (Fig. 3). 
6 

The surface of each block consisted of sampling units of 0.5 by 0.5 m quadrats. Two 

sampling units per block were chosen at random without replacement. For a 100 m beach 

subsegment, a total of 96 quadrats were chosen. The surface of each randomly selected sampling 

unit was scrutinized for presence of oil, and next excavated to a maximum depth of 0.5 m, or 

(often) less if bedrock was present. The resulting pit was named the origin pit, to distinguish it 

from any succeeding pits, called adaptive pits. Any subsurface oil found in the origin pit was 

visually classified by increasing quantity: oil film (OF), light oil residue (LOR), medium oil 

residue (MOR), and heavy oil residue (HOR). Digging the origin pit could disturb the surface of 

nearby quadrats, so surface oil observations were restricted to the origin quadrats. For subsurface 

oil, no confusion occurred between oil present before sampling and that recently disturbed by 

digging pits. If subsurface oil was found in the origin pit, a neighborhood of four bordering 

adaptive pits, each of the same dimensions as the origin pit, were dug around it–above, below, to 

the right and left. When oil was present in one or more of these adaptive pits, additional 

neighborhoods of adaptive pits were dug around each. Eventually, the entire patch of subsurface 

oil was uncovered, which could extend even beyond the block of the origin pit. The record for 

each randomly selected sampling unit indicated the presence or absence of surface oil, as well as 

the visual index of amount of subsurface oil, and the associated numbers of succeeding adaptive 

pits (Fig. 4). 

The information about oil from the pits was used to estimate the area occupied by 

subsurface oil per stratum by three modes–Stratified Adaptive (Adaptive), Stratified Random 

Sampling (SRS), and Observed. The Adaptive estimate for each stratum used both the randomly 

selected and adaptive pits, first expanding oiled area within sampled subsegments for the 
7

unsampled portion, and then expanding estimated oiled areas in sampled subsegments for 

unsampled subsegments in the stratum. The SRS estimate differs from the Adaptive estimate in 

that only the randomly selected origin pits were used in the calculations, ignoring the adaptive 

pits, and the expansion within sampled subsegments differs as a result. This allowed us to 

compare the efficiency of the Adaptive and SRS estimators. Neither the Adaptive nor the SRS 

estimator of oiled area per stratum are theoretically biased. The Observed estimate used the oil 

patches found by the random and adaptive pits without expansion for the unsampled portion 

within subsegments, but with expansion for unsampled subsegments in the stratum. The 

Observed estimator of oiled area per stratum is biased low, and no variance is computed for 

uncertainty at the beach segment level. Because the adaptive pits were not scored for surface oil, 

neither an Adaptive estimate nor an Observed estimate was available for surface oil, only an SRS 

estimate based on the randomly selected origin pits. Because information about the weight of oil 

was derived from subsurface observations, and not surface observations, only subsurface oil 

weight could be estimated using the Adaptive, SRS, and Observed approaches. 

Gravimetric Samples 

Gravimetric samples of oiled beach material were collected in order to calibrate the visual 

oiling classification to the physical amounts of oil present in the pits. A total of 100 pits were 

sampled on a schedule of 0-5 pits per day throughout the field season in order to obtain broad 

coverage of Prince William Sound beaches for the four oiling categories (OF, LOR, MOR, 

HOR). These samples were not chosen at random, but rather on systematic basis due to the 

uncertainty of obtaining sufficient replicates of each oiling category. Each gravimetric sample 
8

was a subsample of the thoroughly mixed material from an oiled pit. The weight of oil present in 

each subsample was determined by chemical extraction and gravimetric measurement, and the 

total weight of oil in the pit was estimated by simple expansion for subsampling. The estimated 

total weight of oil per unit surface area for the pit was obtained from the pit surface area of 0.25 

m2. The oiling category OF was subsequently dropped from the analysis as only two gravimetric 

samples were taken in the field. Weight of oil for the OF category was assumed to equal zero. 

Estimation Formulas 

Oiled-area for a Selected Beach Subsegment 

A stratified estimator of modified Horvitz-Thompson type (Chapter 26, in Thompson 

1992) provides each subsegment Adaptive estimate of oiled area, y$, and its variance, 

H H H y y I I ⎛ p ⎞

y$ = ∑ y Ii i, var( y$)= ∑ ∑ i j i j

⎜

⎜ ij 

− 1⎟

⎟ ,

i=1 pi i=1 j=1 pij ⎝ p pi j ⎠

where H oil patches occur in the entire subsegment, 

yi is the area of the ith patch, 

Ii is a 0-1 indicator for the intersection of the ith patch by the initial pairs of random 

quadrats from the L blocks comprising the surface of the beach subsegment, 

pi is the probability of this intersection, and 

pij is the probability of intersection of both the ith and jth patches by the initial pairs of 

random quadrats. 
9 

The probability of inclusion of the ith patch in the initial random quadrats is 

ki ⎛

⎜

⎝

⎞

⎟

⎠

N

2

k ⎞

⎟

⎠

, L 

pi = −1 ∏k =1 

where 

⎛

⎜

⎝

N − x k 

2

Nk quadrats comprise the surface of the kth block, and

xki quadrats in the kth block intersect the ith patch.

The probability that the initial random quadrats intersect both the ith and jth patches is 

⎛ N − − xki x ⎞ ⎛ N ⎞ L k kj k 

ij 1 1( p = − − p i ) − (1 − p j ) + ∏k =1 

⎜

⎝ ⎟

⎠ ⎜

⎝ ⎟⎠ . 2 2

A stratified random sample (SRS) estimates of oiled areas in each subsegment, y$ and its 

variance var( ) y$ were computed as follows: 

( p$ k (1− p$ k )) L L 

∑ N p $ ( ) k k y$ = , var( y$)= ∑ N N − n , k k k 

k =1 k =1 n k − 1

where Nk is the total number of quadrats in block k, nk (= 2) is the number of random quadrats 

sampled in block k, p$ k is the observed proportion (0, 0.5, or 1) of random quadrats in block k 

that were oiled. 
10 

Later, stratum and region totals for oiled area and weight were estimated using the 

Adaptive and SRS subsergment area estimates as well as the Observed subsegment areas. 

Oiled-area Estimation for Strata 

Prince William Sound beach subsegments were partitioned into three oiling categories 

and two length categories. Consider an oiling category in which N1 subsegments were 100 m and 

N2, shorter lengths, denoted by L1, L2,ÿ, LN2. Let n1 , the 100 m subsegments to be sampled be 

drawn at random without replacement from the N1 available, and n2 subsegments of the second 

category be drawn with ppl sampling from the N2 available. To do ppl sampling, n2 units were 

drawn with replacement and probabilities pi = Li/(L1+ÿ+LN2) = Li/L. Adaptive stratified sampling 

(Thompson 1992) was applied to each selected beach subsegment for estimation of oiled area 

and oil weight. Let y11,y12,ÿ,y1n1 be the unobserved oiled areas in the selected beach subsegments 

of the first stratum, and let y^11,y^12,ÿ, y^1n1 be the corresponding estimated oiled areas. Let y21,y22,ÿ, 

y2n2 and y^21,y^22,ÿ, y^2n2 be the unobserved and estimated oiled areas for the ppl sample from the 

second stratum. 

N1 

The unobserved total oiled area from the first stratum is ∑ y1i = T1 . If the y1i were 

i=1 

observable, an appropriate estimate of total oiled area would be T$ = N y , where y1 is the 1 1 1 

2 

sample average of the y1i, and its estimated variance would be var(T$ ) = N (N − n ) s1 , 1 1 1 1 n1 

where s1

2

 is the sample variance of the y1i (Thompson 1992; p. 103). Because the y1i are not 

observable and must be estimated, the estimate becomes 

⎛ ⎞ n1 

⎜ ∑

⎜

⎜

⎜

⎝

⎟

⎟

⎟

⎟

⎠

y$1i 

i 1 T$ N y ~ . (1) =

1 = 1 = N1 n1 
11 

The corresponding estimated variance of T

$

1 is 

2 

1 N ⎛

y$1 ( )⎞

⎟

⎠

n1 s ∑1 

n1 n1 

V T $( $ ) = N ( N − n ) 1 1 1 1 ⎜

⎝ + var , (2) i 

i =1

where s1

2

 is the sample variance of the y^1i, and var( y$1i ) is the estimated error variance for the ith 

beach subsegment from stratified adaptive sampling. The first term of the estimated variance 

represents the variation among 100 m beach subsegments, and the second term is the 

contribution due to estimation error of the selected subsegments. The variance estimator is 

obtained from Equation 6 in Thompson (1992; p. 129), but with var( y$1i) from stratified adaptive 

sampling substituted for the variance of the estimated total oiled area had simple random 

sampling of quadrats within the beach subsegment been used. Thompson’s Equation 6 is an 

unbiased estimate of the variance for a population total obtained by simple random sampling at 

each of two stages, primary units and secondary subunits (see also Appendix I). N2 

The unobserved total area from the second stratum is ∑ y2i = T2 . Were the y2i 

i=1 

observable and drawn with replacement and probability equal to Li/L (L = ELi), the HansenL n2 

∑ y2i Hurwitz estimator in Thompson (1992; p. 47) for T2 would beT

$

2 = with each value, 

n2 i =1 Li

y2i, used in the sum as often as the ith beach subsegment was selected. The estimated variance of 

the Hansen-Hurwitz estimator would be 

2 

y2i − T

$

2 $( $ V T ) 2 = L2 n2 

∑ ⎛ ⎞ ⎜

⎝ ⎟

⎠ . nn ( − 1) 2 2 i =1 Li L
12 

Because the y2i are not observable and must be estimated,

L n y$2i 2 

T

$

2 = ∑ . (3)

n2 i =1 Li

The corresponding estimated variance of T

$

2 is 

2 

$( $ V T ) 2 = L2 

nn ( − 1) 2 2 

⎛ y$2i 

Li 

− T$

2 

L

⎞ n2 

∑ ⎜

⎝ ⎟⎠ , (4)

i =1

where values of the y^2i correspond to the estimated total in the ith beach subsegment selected 

(repeated selections of a particular subsegment result in differing estimates included in the sum). 

This variance formula is described in Thompson (1992; pg. 132). 

Sampling for oiled-area estimation is done independently between subsegment-length 

strata and among the three oiling categories. Therefore, estimates of total oiled area in Prince 

William Sound is estimated as the sum of the six oiled-area estimates (three oiling categories × 

two subsegment-length categories) and the variance of this sum is the sum of the six individual 

variances. Estimates of oil weight are not done independently among subsegment-length strata 

and oiling categories, as we see next. 

Oil-weight Estimation 

In order to estimate the weight of oil on the beaches, a discrete visual scale having M (= 

4) levels of oil contamination was defined. Consider a particular subsegment-length stratum of an 

oiling category. Denote its total oiled areas belonging to the M visual categories of oil 
13

contamination by AN = (A1,ÿ,AM), and let :wm be the average weight of oil per unit area of Am. 

Then the weight of oil on its beaches is the total of visual-category products of the unobserved 

oiled areas and average weights of oil per unit area, 

M 

W = ∑ Am μ wm . m=1 

Estimates of areas in a stratum belonging to these visual categories are derived from the 

field sampling data. The field sampling for each subsegment allowed estimation of both the total 

oiled subsurface area and the proportions of that area belonging to the visual categories. Recall 

that in regular field sampling, the beach surface of each selected 100 m subsegment was divided 

into 48 blocks, and each member of a pair of random sampling units from each block was 

examined for surface and subsurface oil. If the subsurface was oiled, the origin pit was classified 

to a visual category of contamination. The beach surface of shorter (< 100 m) subsegments was 

divided into fewer blocks, but again each member of a pair of random origin pits from each block 

was classified to a visual category of subsurface oiling. The number of origin pits in the oiled 

area of a sampled beach subsegment was random, depended on the size of the oiled area, and 

actually ranged from 1 to 31, out of a possible 96, for 100 m subsegments, and 1 to 35 for 

< 100 m subsegments. Later, these subsegment counts that classified to the M visual categories 

are modeled with the multinomial probability function when estimating the visual-category 

proportions that compose the subsurface oiled area of a sampled beach subsegment. 

Estimates of the average weight of oil per unit area within visual oiling categories were 

obtained from the gravimetric samples of material in oiled origin pits. Denote by nw1, nw2, ..., 
14

nwM the numbers of pits from each of the M visual categories that were processed for weight of 

oil. The underlying means and variances for the weight of oil per unit area in the M visual 

categories, :w1, :w2, ..., :wM, and Fw1

2

, Fw2

2

, ..., FwM

2

 are estimated from the gravimetric samples 

by sample averages of their weights per unit area, 1 , 2 , wM ww ,K , and associated sample 

variances, sw1

2

,sw2

2

,..., swM

2

, respectively. 

The oil weight estimation will be described in reverse order for the two categories of 

beach subsegments, beginning with <100 m subsegments and finishing with 100 m subsegments. 

< 100 m Beach Subsegments 

The total weight of spill oil in the second stratum comprised of < 100 m beach 

subsegments within an oiling category will be denoted by W2. This weight can be written as 

M 

W2 = ∑ Am μ wm , m=1 

where Am is the oiled area comprised of visual category m, and :wm is the average weight of oil 

per unit area of Am. Each sampled beach subsegment provides an estimate of AN=(A1,ÿ, AM), the 

visual-category areas of the entire stratum. If a single beach subsegment, say the ith subsegment, 

were drawn, the Hansen-Hurwitz estimate of A would be 

⎛ L $ $ L y pi 1i 

i 

L 

y p $ $i 2i 

⎞ ⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

A$ i = Li ,

M

L 

y p $ $ Li 

i Mi ⎝ ⎠
15

where y$i is the estimate of the oiled area from adaptive sampling, and p$i = ( p$ , , K p$ )′ is the 1i Mi 

vector of estimated proportions of the oiled area belonging to the M visual categories in the 

beach subsegment. The beach subsegments were drawn independently with replacement, and if a 

beach subsegment was drawn more than once, its oiled area and area proportions of the M visual 

levels were estimated repeatedly and independently of previous draws. As a result, the n2 

estimates, A$ i , i = 1,ÿ,n2, are independent and identically distributed random vectors from an 

underlying multivariate sampling probability distribution for estimated stratum totals of the M 

visual categories of oiled areas. Their average, 

⎛ A1 ⎞ ⎛ p$i1 ⎞ ⎜

⎜

⎜

⎜

⎝

⎟

⎟

⎟

⎟

⎠

Ly$i 

Li 

⎜

⎜

⎜

⎜

⎝

⎟

⎟

⎟

⎟

⎠

A2 

M

AM 

1 

n i=1 i=1

2 n2 

n 1 2 n2 p$i

A$ 2A = = ∑ = ∑ , (5) i M

p$iM 

is an unbiased estimate of the visual oiled areas in the stratum, and it has unbiased variancecovariance matrix estimate, 

s

2 

s K s A1 A1 , A2 A1 , AM 

− s

2 

K s 

⎛ ⎞ ⎜

⎜

⎜

⎜

⎟

⎟

⎟

⎟

1 = ⋅ A2 A2 A

S A = (sAi A ) , M= − − O M n n( 2 −1) , j 

2 

− − − s

2 

AM ⎝ ⎠

( A 1 ) ( A 1 )( A 2 ) ( A 1 )( A M ) ⎛ n n n

⎞ 2 2 2

∑ $

i1 − A ∑ $i1 − A $i 2 − A K∑ $i1 − A $iM − A

(6) = i=1 i=1 

( A 2 ) ( A A A)(

2 

− AM 

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

) .

n 2 2 n2

− −−− − − $ − A $ − $ ∑ i 2 K∑ i 2 2 iM

i=1 i=1 

M M M

( iM M ) n2 

− −−− − − −−− − − K ∑ A$ − A

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝ i=1 

i 

2 

1 
16 

The unknown weight of oil in the stratum, W2, is estimated as 

M 

W$

2 = A w , (7)

and it has variance (See Appendix II) 

∑

m=1 m m 

2 2

σ wi 2

( ( σ wi ⎧

⎨

⎩

Ai 

2 ⎫

⎬

⎭

M M 

( $ V W )2 = μ wiV Ai ) V Ai ) 2 μ μwi Cov ( Ai Aj ∑ + + + ∑ ∑ , ) . wj i=1 n n i =1 j i < wi wi 

This variance is estimated unbiasedly by 

⎧

⎨

⎩

Ai 

2 ⎫

⎬

⎭

2 2 M M swi swi V W$( $

2 ) = 22 2

∑ + ws + s + 2∑ ∑ ww s i j , (8) i Ai Ai i A A, j

n nwi 

i=1 i =1 j i < wi 

where sAi 

2 the estimated variance of Ai , and sA A i j , is the estimated covariance between Ai and 

Aj . 

100 m Beach Subsegments 

The total weight of spill oil in the first subsegment-length stratum, comprised of 100 m 

beach subsegments, will be denoted by W1. Let ai=(ai1, ai2,ÿ, aiM)N denote the oiled areas 

composing the visual categories in beach subsegment i, and AN=(A1, ÿ, AM) denote the total areas 

by visual category in the entire stratum. For a sample of n1 beach subsegments, the Mdimensional visual area arrays, a1, a2, ÿ, an1 are estimated by adaptive sampling. The visual area 
17

estimates are denoted by the vectors, a a $ , , 1 , $2 K a$n , where a$i = $ p$i yi . Recall that y$ is the estimated i 1

total oiled area from adaptive sampling in the ith beach subsegment, and p$i is the associated 

vector of estimated proportions of the oiled area belonging to the M visual categories. The visualcategory areas in the stratum are estimated by expanding the sample average as 

′ ⎛ ⎞ n1 n1 n1 

∑ a$i1 ∑ a$ ⎜ i2 ∑ ⎜

⎜

⎜

⎝

⎟

⎟

⎟

⎟

⎠

a$iM 

i 1 i 1 i 1 A$ = N = = = ~ ⋅ , , , K = N a ′ (9) 1 1 n1 n1 n1 

If the areas within each beach subsegment could have been directly observed rather than 

estimated as was necessary, the variance-covariance matrix of these area estimates would be 

V

VA$ = N1 ( N1 − n1 ) a , n1

where Va is the population variance-covariance matrix of the arrays, a1, a2, ÿ, aN1. The estimate 

of this matrix would be 

Sa V$

a$ = N1 (N1 − n1 ) , n1

where Sa is the sample variance-covariance matrix with elements 

1 n1 

∑ (a ki − a i)(a kj − a j ). k =1 

s = a, ,ij n1 − 1
18 

Here ai = 1 ∑

n1 

aki is the sample average area of visual category i. n1 k =1 

Because the ai were not observed and had to be estimated, the variance-covariance matrix 

of A$ includes a term for this estimation error and is 

VA$ = N1 ( N1 − n1 ) 

Va +N1 ∑

N1 

V$ − . a a n1 n1 k =1 k k 

Here V$ is the variance-covariance matrix of estimation errors in beach subsegment k (see ak −ak

Appendix III), 

2 2 2 ′ V = ( y + σ )⋅V + σ (p p ) . a$ a k − k k y$ k p p $ k − k y$ k k k 

The matrix, Vp$k − pk, is the variance-covariance matrix of the estimated visual category 

proportions in the kth beach subsegment from multinomial sampling of origin pits in the oiled 

area. 

The jth diagonal element of the matrix, Va$k −ak , can be written as 

2 2 kj kj 2 2 

aa = yk v $ − , ,j j ( + σ y $ ) + σ y ⋅

p (1 

h 

−

k

p ) 

$ pkj , k k k k 
19 

where 

pkj is the proportion of the oiled area in the kth beach subsegment belonging to the jth 

visual category of oiling, 

yk is the oiled area of subsegment k, 

σ y

2 

$k 

is the variance of the estimated oiled area in subsegment k, and 

hk is the number of origin pits that were found to be oiled and were classified to the 

several visual categories. 

The off-diagonal element in the ith row and jth column (i ... j) is 

p p 2 2 ki kj 2 v $ = − ( y + σ )⋅ hk 

+ σ pp . ak −ak , ,i j k y$k y$k ki kj 

An unbiased estimate of Va$k −ak is denoted by Sa$k −ak with diagonal elements 

−1 −1 ⎡ ⎤ ⎡ ⎤ ⎡ yk 

2 ⎤ ⎛

⎜

⎝

1− 1

hk 

⎞

⎟

⎠

⎛

⎜

⎝

1− 1

hk 

⎞

⎟

⎠

$ σ$ 2 ⎢

⎢⎣

⎥

⎥⎦

⎢

⎢⎣

⎥

⎥⎦

p$ kj (1− p$ kj ) − p$ kj (1− p$ kj + p$ ) ⎢

⎣ ⎥

⎦

s = , a$k −ak , ,jj y$ kj hk k 

and off-diagonal elements, 

−1 ⎡ ⎤ 2 ⎡ ⎤ ⎛

⎜

⎝

1− ⎞

⎟

⎠

1

hk 

y$ k + σ$ 2 ⎢

⎢⎣

⎥

⎥⎦

= − p p $ $ ⎢ ki ⎣ ⎥

⎦

s . a$k −ak , ,ij y$ kj hk k 
20 

The estimate of V $ is A 

S N n1 

∑1 

n1 n1 k 1=

a$ S N (N − S 1 = n ) + −a (10) A 1 1 k $ a$ , k 

where

Sa$ is the sample variance-covariance matrix of the a$ k , k = 1 ,ÿ, n1, and 

Sa$ k −ak is the estimated variance-covariance matrix of estimation errors in the kth 

sampled subsegment. 

The unknown weight of oil in the stratum, W1, is estimated as 

M 

W$ = ∑ A$ w , (11) 1 m m m=1 

and it has a variance of 

2 2

σ wi 2

( ( σ wi 

wi wi 

⎧

⎨

⎩

Ai 

2 ⎫

⎬

⎭

M M 

( $ V W )1 = μ wiVA i V A i μ μwi wjCov A ) ) 2 ( i Aj ∑ + + + ∑ ∑ , ) . i =1 n n i =1 j i <

This variance is estimated unbiasedly by 

$( $ V W )1 

M ⎧

⎨

⎩

A$ 2 ⎫

⎬

⎭

⎛ ⎞ 2 2 M 

2∑ ∑ swi s 2 wi − 2

∑ + ⎜

⎝

wi 

n

⎟

⎠

sAi = + w w s i j i A A, j 

, (12) i 

i=1 n i =1 j i < wi wi 
21

where sA$

i

2 is the estimated variance of A$

i , and s$ $ is the estimated covariance between A$

i and A A, i j

A$

j . 

Total Weight of Spill Oil Remaining in Prince William Sound 

The total subsurface oiled area in Prince William Sound by visual category equals the 

sum of individual components, 

3 2 

A••m = Aijm ∑ ∑ , m = 12 , , K, M , i=1 j=1 

where Aijm is the total oiled area of visual category m in the jth subsegment-length category (j = 1,

or 100 m, j = 2, or < 100 m) of the ith oiling category (i = 1, 2, or 3). 

The total weight of oil remaining in Prince William Sound is

M 

W = ∑ A μ . ••m wm 

m=1 

The visual category totals, ACCm, are estimated by replacing each Aijm by its estimate 

(Equations 5 and 9) in this equation, and the total weight of oil remaining in Prince William 

Sound is estimated by 

M 

W$ = ∑ A$

••m wm . (13) m=1 
•• •• •• ••

•• ••

••

•• ••

22 

The variance of the estimated total weight of oil remaining is 

m )

σ

n 

2 

wm 

wm 

⎫

⎬

⎭

+ 2

M M σ2 

wm ⎧

⎨

⎩

VW ( $ ) = ∑ ∑ ∑ 2 2 ( $ ( VA $ ) + VA Cov ( A$ , A$ A•• + μ μ μ wm j ) . m wm m wj m n m=1 wm m=1 j m<

This variance is estimated unbiasedly by 

M 2 2 M ⎧

⎨

⎩

⎫

⎬

⎭

s ( ( s 22 2

ws s ( ∑ ∑ ∑ 2 $( $ VW ) = A$ wm wm + + + 2 w w s (14) $ $ , m n m A$

••m A$••m n m j A ••m , A j m=1 wm wm m=1 j m<

where 

s

(

A 

2 

$

••1 

s

(

A$

••1 , A$••2 

K s

(

A$

••1 , A$•• M 

s

(

A$ A$ s

(

A 

2 

$ K s

(

A$ A$

⎛ ⎞

⎟

⎟

⎟

⎟

⎟

⎠

⎜

⎜

= ⎜

A$

M O

⎜ (

s KK (s

⎜

⎜

⎝ ⎟

is the estimated variance-covariance matrix for the visual area array comprised of elements, 

A$

••m , m = 1,K, M . The elements of this covariance array are obtained by summing 

corresponding elements of the estimated variance-covariance matrices of visual areas,

S A or S A$ (Equations 6 or 10), of the six combinations of subsegment-length and oiling 

categories. 

(

S , , ••2 ••1 2 ••2 M 

2 

A$ A$ A$ M , ••1 M 
23 

The formulas for estimates and their standard errors (square root of variances) for oiled 

areas and weight of oil remaining in Prince William Sound are complete. Approximate 95% 

confidence intervals can be computed for any estimate by subtracting (lower limit) and adding 

(upper limit) 2 standard errors to the point estimate. If the distribution of the estimator is not 

approximately normal, confidence intervals are better computed by the bootstrap method. These 

bootstrap estimates of standard errors and confidence intervals were computed as described the 

following section. 

Bootstrapping for Determination of Precision of Estimates 

Adaptive, SRS, and Observed estimates of oiled areas and weights are computed by 

bootstrapping methods. As before, the estimation is described in reverse order for the two 

categories of beach subsegments, beginning with < 100 m subsegments and finishing with 100 m 

subsegments The average, standard deviation, and lower 2.5 percentile and upper 97.5 percentile 

of the empirical distributions are tabulated and reported. The percentiles provide a 95% 

confidence interval. 

Oiled Area, <100 m Subsegments 

The total oiled area in each of the three oiling categories (heavy oiling, 1990-1993; 

medium oiling, 1990-1993; and heavy oiling, 1989) has been estimated by the Hansen-Hurwitz 

formula from oiled areas, Observed, SRS, or Adaptive found in n2 ppl randomly selected beach 

subsegments of N2 available (n2 and N2 vary among oiling categories). Let 
24 

L 

zi = yi i = , , K,n2 $ , 12 , Li 

denote the estimate of the stratum total oiled-area from the ith subsegment. The Hansen-Hurwitz 

estimate is the average of the zi. The zi of each stratum are independent and identically distributed 

random variables, so a standard bootstrapping method is used to draw samples by which to 

estimate precision of the estimated total oiled area for a stratum: 1,000 resamples of size n2 are 

drawn with replacement from the n2 original z-values of the sample of beach subsegments. 

* * * Denote these values for the bth resample as b1 , , zz b K, z . Notice that each resample is 2 bn 2 

composed of the original z-values of the sampled subsegments, each repeated a random number 

of times from 0 to n2. The stratum total oiled-area estimate computed by the Hansen-Hurwitz 

formula from subsegments of the bth resample, say T2b

*

, b = 1,2,...,1,000, equals the bth 

resample average of its z

*

-values. The average, standard deviation, and lower 2.5 percentile and 

upper 97.5 percentile of the empirical distribution of the 1000 T2b

*

 are recorded for Adaptive, 

SRS, and Observed oiled areas. 

Oiled Area, 100 m Subsegments 

The total oiled area in each of three strata (heavy oiling, 1990-1993; medium oiling, 

1990-1993; and heavy oiling, 1989) has been estimated by simple expansion of average oiled 

areas among the n1 sampled subsegments, either Observed, SRS, or Adaptive. To evaluate the 

precision of each stratum estimate of total oiled area, 1,000 bootstrap resamples of n1 

subsegments are drawn. The populations of subsegments and sampling method for generating 

these bootstrap samples depends on the stratum sampling fraction, n1/N1. If n1/N1 is small 
25

(< 0.05), dependence caused by sampling without replacement from the finite population of size 

N1 is ignored, and standard bootstrapping methods for independent and identically distributed 

random variables apply. The bootstrap samples are drawn with replacement from the n1 

subsegments originally sampled. If n1/N1 is larger ($ 0.05), the dependence is not ignored, and 

bootstrapping methods for finite populations apply. This method used approximations suggested 

by Gross (1980), as described by Booth et al. (1994). The bootstrap samples are drawn without 

replacement from a population of [N1/n1] +1 copies of the original n1 subsegments, where the 

expression, [A], denotes the integer part of the argument. Notice that any resample is composed of 

the estimated oiled areas of the original sampled subsegments, each occurring between 0 and n1 

times if n1/N1 is small, or between 0 and [N1/n1] +1 times if n1/N1 is large. Denote the estimated 

oiled areas of the subsegments in the bth bootstrap sample as b1 b2 ,, $bn 1 . Let T1b

* y y $ , b = *

, $ * K y 

* 

1,2,...,1,000, be the stratum total oiled-area estimates computed by expansion of the average 

oiled areas among subsegments for the ith resample, 

⎛ n1⎞ y$bi 

*

⎜ ∑

⎜

⎜

⎝

⎟

⎟

⎟

⎠

T* N y ~* 

1b 1 b N i=1 = = . 1 

n1 

The average, standard deviation, and lower 2.5 percentile and upper 97.5 percentile of the 

empirical distribution of the 1,000 T1b

*

 are recorded for Adaptive, SRS, and Observed oiled areas. 

Visual-category Composition of Oiled Areas, < 100 m or 100 m Subsegments 

Each subsegment sampled and found oiled during the survey has an estimated visualcategory composition, p$ , from the h random quadrats in its oiled area. Specifically, if the counts 
26

of quadrats in the four visual categories were h = ( h1,h2,h3, h4), the visual-category composition of 

the oiled area is estimated as 

4 

p $ = (hh K,h / h , h = . 1 /, 4 ) ∑ hi i=1 

A total of 1,000 bootstrap samples of h = ( h1,h2,h3, h4), are obtained by one of two 

procedures, depending on the apparent sampling fraction. If the sampling fraction, h y$, is less 

than 0.05, a sample of h quadrats is drawn with replacement from the original h quadrats 

observed. If the sampling fraction exceeds 0.05, a sample of size h is drawn without replacement 

from a population composed of [ y h $ ] + 1copies of the original h random quadrats in the oiled

area, where [@] truncates the number to its integer part. In either case, denote the bootstrap sample 

by h* = (h1

*

,ÿ, h4

*

). Then the corresponding estimated visual-category composition is 

*

( $ *, $ *,, $ *) * * p$ = pp K p = (h / h,K,h / h ). 1 2 4 1 4 

Average Weight of Oil per Quadrat of the Visual Categories 

The n gravimetric-extraction samples of visual category m are sampled with replacement n 

w * times to provide averages, bm , b = 1 2, , K, 1 , 000 ;m = 1,2,3, 4 . 

Oil Weight for Strata with Subsegments < 100 m 

The algorithm for computing bootstrapped values of the total oil weight for an oiled 

category of <100 m subsegments is as follows: 
27 

1. Set b=1 

* * * 2. Let zz K, z denote the bth bootstrap resample of previously calculated z-values for a , , b1 b 2 bn 2 

stratum of < 100 m subsegments. Corresponding to these z

*

-values are the particular beach 

subsegments from which they were computed. For these subsegments, generate bootstrap samples 

for the visual-category compositions, $ *

, ,p$ * p as calculated above. b1 K bn2 

3. Compute the estimated visual-category oiled areas in the stratum by 

⎛ ⎞ ⎛ ⎞ * * Ab p$ bi ⎜

⎜

⎜

⎟

⎟

⎟ =

⎜

⎜

⎜

⎟

⎟

⎟

,1 ,1

1 

n2 i=1 

n2 

∑ * * Ab = M z M . bi 

* * Ab p$ ⎝ , 4 ⎠ ⎝ bi,4 ⎠

4 

W$

2 A * * * 4. Compute the total weight of oil in the stratum by = ∑ w using the bootstrapped ,b b,m b,m m=1 

samples of average weight per quadrat as calculated above. 

5. If b < 1,000, set b = b + 1 and go to step 2. Otherwise, stop bootstrap sampling. 

Oil Weight for Strata with Subsegments of 100 m 

The algorithm for computing bootstrapped values of the total oil weight for an oiling 

category of 100 m subsegments is as follows: 

1. Set b =1. 

* * * 2. Let y y $ ,, $ b1 , $b 2 K ybn 1 denote the bth bootstrap resample of oiled-area estimates for a stratum of 

100 m subsegments previously calculated above. Corresponding to these oiled-area estimates are 

the particular beach subsegments from which they were computed. For these subsegments, 
28

generate bootstrap samples for the visual-category compositions p$ * , , K p$ * as described above.

b1 bn 1 

3. Compute the visual-category areas of each beach subsegment in the bth bootstrap resample as

* *

⎛a$bi1 ⎞ ⎛ p$ bi1 ⎞ ⎜

⎜

⎜

⎝

⎟

⎟

⎟

⎠

= ⎜

⎜

⎜

⎝

⎟

⎟

⎟

⎠

* * a$bi = M y$bi M , i=1,2,ÿ,n1. 

* * 

a$ p$ bi 4 bi 4 

4. Compute the estimate of total oiled area by visual-category in the stratum for the bth bootstrap 

sample as 

⎛ ⎞ n1 

a$

* ∑ bi1 *

A$

b ⎛ ⎞ ⎟

⎟

⎟

⎟

⎟

1 i=1

⎜

⎜

⎜

⎝

⎟

⎟

⎟

⎠

N1 

n1 

*

A$

b = M = M .

A$ * n1

⎜

⎜

⎜

⎜

⎜

⎝ a$

* ∑ bi 4 b4

⎠ i=1 

4 

W$ A$ * * * 5. Compute the total weight of oil in the stratum by ∑ wb m = using the bootstrapped 1, b b,m m=1 

samples of average weight per quadrat as previously calculated. 

, 

6. If b < 1,000, set b = b + 1 and go to step 2. Otherwise, stop bootstrap sampling. 

The algorithm above for total oiled area and oil weight apply to a single stratum. For 

Prince William Sound totals of area estimates, the bth area estimates, T*

1b and T*2b, are summed 

over the strata to provide the bth bootstrap estimate of the Prince William Sound total oiled area. 

* For Prince William Sound totals of weight estimates, the bth estimates, W$

1,b and W$

2

*

,b , are summed 
29

over the strata to provide the bth bootstrap estimate of the Prince William Sound total oil resulting 

in a distribution of 1,000 bootstrap estimates. 

RESULTS AND DISCUSSION 

Sampling was concentrated (17% - 20% sampling fraction, by number of subsegments, or 

20% - 27% by length) on beaches for which the most recent surveys found heavy oiling (lines 1 

and 2, Table 1). Lower intensity (3% - 5% sampling fraction, by number of subsegments, or 3% ­

8%, by length) was directed to beaches found to have medium oiling during these surveys (lines 3 

and 4, Table 2), and yet lower intensity (2% sampling fraction, by number of subsegments, and 

2%, by length) was applied to beaches found with heavy oiling in earlier surveys, but less than 

medium oiling more recently. 

Comparison between subsurface oiled area estimates within sampled subsegments (strata 

combined) for the SRS and Adaptive modes (Fig. 5) shows good agreement as expected, but with 

several material discrepancies for subsegments with larger oiled areas. In particular, SRS 

underestimated the known minimum oiled areas on four subsegments (Fig. 6) in contrast to 

adaptive estimation (Fig. 7). In general, the estimated standard errors of oiled areas in the sampled 

subsegments by the two modes reflect the reduced uncertainty by including the adaptive pits 

(Fig. 8). 

Estimates of the total subsurface oiled area per stratum and measures of their precision 

were computed for Adaptive, SRS, and Observed modes, either from analytical formulas or by 

bootstrap resampling (Table 2). As expected, Observed estimates of oiled quadrats within 
30

subsegments, when extrapolated to stratum totals, provide minimal, biased values. Adaptive and 

SRS estimates are in good agreement for the more heavily sampled strata 1 and 2, and in fair 

agreement as sampling intensity declines in strata 3, 4, and 5. Analytical and bootstrap 

computations are generally in good agreement, but only the bootstrap confidence intervals 

appropriately shift to accommodate the skewed distributions. Considering the bootstrap 

computations for the Adaptive estimates as most trustworthy, strata 1 and 2 had considerable oiled 

area, 13,000 m2 (4,400 - 24,600 m2), and 22,000 m2 (10,300 - 36,000 m2), respectively. Stratum 4 

may also have had substantial oiled area, estimated imprecisely at 27,000 m2 (2,400 - 63,900 m2). 

The other strata, 3, 5, and 6, had less oiled area, also imprecisely determined as 6,000 m2 (900 ­

13,800 m2), 3,000 m2 (0 - 9,700 m2), and 0 m2 (none observed), respectively. Although the 

separate strata provide relatively imprecise subsurface oiled area estimates, the grand sum for 

Prince William Sound is reasonably precise (Table 3), with Adaptive estimates from the bootstrap 

computations equaling 71,000 m2 (37,700 - 113,200 m2). The differences in grand sum of 

subsurface oiled area for Prince William Sound, whether Adaptive or SRS estimates, and 

analytical or bootstrap computations, are relatively less concerning as well (Table 3). 

Estimates of the total surface oiled area per stratum and measures of their precision were 

computed for stratified random sampling (SRS), either from analytical formulas or by bootstrap 

resampling (Table 4). Analytical and bootstrap computations are generally in good agreement, but 

only the bootstrap confidence intervals accommodate the skew distributions reasonably. 

Considering the bootstrap computations as most trustworthy, the total surface oiled area was 

estimated for strata 1, 2, and 3 as 8,000 m2 (3,300 - 12,700 m2), 12,000 m2 (6,300 - 19,000 m2), 

and 3,000 m2 (800 - 4,800 m2), respectively. The corresponding estimate for stratum 4 was very 
31

imprecise at 19,000 m2 (1,600 - 46,500 m2). No surface oil was found in strata 5 and 6. The grand 

sum for Prince William Sound is 41,000 m2 (20,700 - 70,500 m2) (Table 5). 

Estimates of the total subsurface oil weight (metric tons (t), or 1,000s kg) per stratum and 

measures of their precision were computed for Adaptive, SRS, and Observed modes, either from 

analytical formulas or by bootstrap resampling (Table 6). Observed estimates of oil weight, when 

extrapolated to stratum totals, provide minimal, biased values. Considering Adaptive estimates 

and bootstrap computations as most reliable, total weights of subsurface oil for strata 1 through 6 

were estimated at 8 t (2.7 - 13.8 t), 18 t (6.3 - 34.7 t), 4 t (0.3 - 10.9 t), 17 t (0.5 - 42.9 t), 4 t (0 ­

11.3 t), and 0 t (none observed), respectively. The grand sum of subsurface oil weight for Prince 

William Sound was 50 t (24.4 - 82.6 t) (Table 7). 

Although the added effort for adaptive pits reduced uncertainty for the oiled areas of 

sampled subsegments, the variation among subsegments within strata greatly increased the 

variances for stratum totals. For example, the variance equation for stratum total oiled area in the 

100 m subsegments (see Equation 2) comprises two terms, the first representing variation among 

subsegments, and the second, estimation error within subsegments. In applying Equation 2 to 

Adaptive estimates of strata 2 and 4, the contribution to variance of stratum total oiled area from 

variation among - subsegments represented more than 99% of total variance. Because the within ­

subsegment variation was negligible as compared to between - subsegment variation, the adaptive 

sampling would have been better implemented by allocating sampling effort among strata from oil 

found in a preliminary exploration, instead of delimiting the oil patches within an oiled beach 

subsegment. For example, stratum 4 contained roughly 40% of all estimated subsurface oil 

(Tables 2 and 3) whereas only 3% of the segments were sampled in that stratum (Table 1). By 
32

implementing an adaptive program on the strata with additional beach subsegments sampled from 

strata where more oil was found, precision in estimation of the amount of Exxon Valdez spill oil 

remaining in Prince William Sound would have been enhanced. Unfortunately, the specific nature 

of the beach sampling permit requirements prevented such adaption to discovery. 

The bootstrap procedure, although computer intensive, provided more realistic measures 

of precision evident from the asymmetric 95% confidence intervals from skewed distributions of 

the bootstrap estimates. The minimal oil estimate should never be less than zero, as occurred with 

the analytical mode, nor less than the amount of oil actually found. The bootstrap calculations 

were more consistent with this principle than the analytical calculations. None of the lower 

bounds from the bootstrap were negative, and only when the oil was very scarce did bootstrap 

resampling fail to produce a lower bound at least as great as the found oil. In implementing the 

bootstrap resampling, the negligible estimation error within 100 m subsegments, but not < 100 m 

subsegments, was omitted from the computations. If sampling effort were better distributed 

among oiling strata in future surveys, the relative importance of estimation error within 

subsegments could increase and necessitate its inclusion in bootstrap calculations. 

Although statistically unbiased within the areas sampled, the estimates of oiled area and 

weight for Prince William Sound are likely biased low nonetheless. The sampling scheme was 

constrained by the tidal nature of Prince William Sound and allowed sampling only at lower tidal 

stages. However, on a number of occasions, our adaptive pits extended below the sampling grid 

which was predetermined at 3.0 m below high tide line. Since the sampling design did not include 

the oil found in this lower intertidal zone, the reported values are minimum estimates. 
33 

CITATIONS 

Booth, J. G., R. W. Butler, and P. Hall. 1994. Bootstrap methods for finite populations. J. Am. 

Statist. Assoc. 89:1282-1289. 

Brodersen, C., J. Short, L. Holland, M. Carls, J. Pella, M. Larsen, and S. Rice. 1999. Evaluation 

of oil removal from beaches 8 years after the Exxon Valdez oil spill, p.325-336. In Proc. 

22nd Arctic & Marine Oilspill Program Technical Seminar, Environment Canada, 

Ottawa. 

Cochran, W. G. 1963. Sampling techniques. Second Edition. John Wiley & Sons, Inc., New 

York, NY. 413 p. 

Efron, B. 1982. The jackknife, the bootstrap, and other resampling methods, Volume 38, 

CBMS-NSF Regional Conference Series in Applied Mathematics, Society for Industrial 

and Applied Mathematics, Philadelphia. 

Finkelstein, K., and E. R. Gundlach. 1981. Method for estimating spilled oil quantity on the 

shoreline. Environ. Sci. Technol. 15: 545-549. 

Gibeaut, J.C., E. Piper, D. Munson, J. Matthews, M. Profita, and C. S. Crosby. 1998. 1993 

shoreline oiling assessment of the Exxon Valdez oil spill. Exxon Valdez Oil Spill 

Restoration Project Final Report (Restoration Project 93038), Alaska Department of 

Environmental Conservation, Juneau, Alaska. 

Goodman, L. A. 1960. On the exact variance of products. J. Am. Statist. Assoc. 55:708-713. 

(Corr. 57: 917). 
34

Gross, S. 1980. Median estimation in sample surveys. Proceedings of the Section on Survey 

Research Methods. American Statistical Association, Alexandria VA. pp. 181-184. 

Koons, C. B., and H. O. Jahns. 1992. The fate of oil from the Exxon Valdez - A Perspective. 

Mar. Tech. Soc. J. 26:61-69. 

Neff, J. M., E. H. Owens, and S. W. Stoker 1996. Shoreline oiling conditions in Prince William 

Sound following the Exxon Valdez oil spill, p. 312-346. In P. G. Wells, J. N. Butler, and 

J.S. Hughs (editors.), Exxon Valdez Oil Spill: Fate and Effects in Alaskan Waters. 

American Society for Testing and Materials Publication 1219, ASTM, Philadelphia, PA. 

Owens, E. H. 1987. SCAT – A ten-year review. p. 337 - 360. in Proc. 22nd Arctic & Marine 

Oilspill Program Technical Seminar, Environment Canada, Ottawa. 

Thompson, S. K. 1992. Sampling. Wiley and Sons, New York, NY. 343 p. 
35 

Table 1.-- Strata sizes, sample sizes, and sampling fractions of the beach subsegments in Prince William Sound classified by oiling 

category, in terms of number and length (m). 

Stratum (h) Segments 

(N

h) 

Sampled Segments (n h) 

Sampling Fraction (%) 

Total Length (m) 

Sampled Length (m) 

Sampling Fraction of Length (%) 

1. 1990-93, heavy, <100 m 133 23 17 5935 1576 27 2. 1990-93, heavy, 100 m 184 37 20 18458 3708 20 3. 1990-93, med, < 100 m 310 16 5 12827 978 8 4. 1990-93, med, 100 m 362 10 3 36303 1000 3 5. 1989, heavy, < 100 m 221 5 2 11294 253 2 6. 1989, heavy, 100 m 317 5 2 31775 500 2 Totals 1,527 96 6 116,592 8,015 7 
36 

Table 2.-- Estimated subsurface oiled area (1,000s m2) per stratum as determined by three modes of expansion 

for unexamined beach area - Adaptive, Stratified random sampling (SRS), and Observed - and precision from analytical and bootstrap calculations.Note that no oil was found in stratum 6. The column “estimate”, refers to point estimate if calculation is analytical, or mean estimate, if bootstrap. Lower and upper limits refer to bounds of the 95% confidence interval (CI). 

Stratum (h) Computation Method Estimate Lower 

95% CI 

Upper 95% CI 

Standard Error 

Coefficient of Variation 

1. 1990-93, heavy, <100 m Analytical Adaptive 13.4 3 23.9 5.35 0.4 

Bootstrap Adaptive 13.3 4.4 24.6 5.18 0.39 Analytical SRS 12.8 3.7 21.9 4.64 0.36 Bootstrap SRS 12.9 4.9 23.2 4.63 0.36 Analytical Observed 10.4 0.4 20.3 5.09 0.49 Bootstrap Observed 10.6 2.5 21.3 4.95 0.47 

2. 1990-93, heavy, 100 m Analytical Adaptive 21.8 9.2 34.5 6.46 0.3 

Bootstrap Adaptive 21.6 10.3 36 6.61 0.31 Analytical SRS 22.3 10.4 34.2 6.05 0.27 Bootstrap SRS 22.6 11.4 36.7 6.48 0.29 Analytical Observed 15.5 3.8 27.2 5.97 0.38 Bootstrap Observed 15.6 4.8 29.3 6.31 0.41 

3. 1990-93, med, < 100 m Analytical Adaptive 5.8 -1.6 13.2 3.78 0.65 
37

Bootstrap Adaptive 5.8 0.9 13.8 3.73 0.64 Analytical SRS 6.9 -1.5 15.4 4.31 0.62 Bootstrap SRS 7.1 0.9 16.6 4.2 0.59 Analytical Observed 1.8 -0.6 4.1 1.2 0.67 Bootstrap Observed 1.7 0.07 4.2 1.14 0.65 

4. 1990-93, med, 100 m Analytical Adaptive 26.6 -7.9 61.1 17.6 0.66 

Bootstrap Adaptive 27.3 2.4 63.9 17.1 0.63 Analytical SRS 29.8 -10.9 70.4 20.7 0.7 Bootstrap SRS 29 2.4 71.4 20.4 0.7 Analytical Observed 18.3 -16.5 53.1 17.8 0.97 Bootstrap Observed 18 0.05 54.4 16.8 0.94 

5. 1989, heavy, < 100 m Analytical Adaptive 3.2 -3.1 9.5 3.22 1 

Bootstrap Adaptive 3.2 0 9.7 2.91 0.91 Analytical SRS 6.6 -6.4 19.7 6.65 1 Bootstrap SRS 6.4 0 19.9 6.25 0.98 Analytical Observed 2.6 -2.5 7.7 0.87 1 Bootstrap Observed 3.2 0 9.4 0.955 0.89 
38 

Table 3.-- Estimated total subsurface oiled area (1,000s m2) remaining in 

Prince William Sound in 2001 as determined by three modes of expansion for unexamined beach area - Adaptive, Stratified random sampling (SRS), and Observed - and precision from analytical and bootstrap calculations. The column “estimate,” refers to point estimate if calculation is analytical, or mean estimate if bootstrap. Lower and upper limits refers to bounds of the 95% confidence interval (CI). 

Computation Method 

Estimate Lower 

95% CI 

Upper 95% CI 

Standard Error 

Coefficient of Variation 

Analytical Adaptive 71 31.5 110.4 20.1 0.28 Bootstrap Adaptive 71.2 37.7 113.2 19.7 0.28 Analytical SRS 78.4 32.4 124.4 23.5 0.3 Bootstrap SRS 78 40.6 127.3 22.8 0.29 Analytical Observed 48.6 10.2 87.1 19.6 0.4 Bootstrap Observed 49.1 19.9 92.7 18.7 0.38 
39 

Table 4.-- Estimated surface oiled area (1,000s m2) per stratum as determined by Stratified random sampling (SRS) 

mode of expansion for unexamined beach area and precision from analytical and bootstrap calculations. Note that no surface oil was found in strata 5 and 6. The column “estimate,” refers to point estimate if calculation is analytical, or mean estimate, if bootstrap. Lower and upper limits refer to bounds of the 95% confidence interval (CI). 

Stratum (h) Computation 

Method 

Estimate Lower 

95% CI 

Upper 95% CI 

Standard Error 

Coefficient of Variation 

1. 1990-93, heavy, <100 m Analytical SRS 8.8 3.9 13.7 2.49 0.28 

Bootstrap SRS 7.6 3.3 12.7 2.44 0.32 

2. 1990-93, heavy, 100 m Analytical SRS 12 5.9 18 3.1 0.26 

Bootstrap SRS 12.1 6.3 19 3.29 0.27 

3. 1990-93, med, < 100 m Analytical SRS 2.7 0.6 4.7 1.05 0.39 

Bootstrap SRS 2.6 0.8 4.8 1.02 0.39 

4. 1990-93, med, 100 m Analytical SRS 19.4 -6.1 45 13 0.67 

Bootstrap SRS 19 1.6 46.5 12.5 0.66 
40 

Table 5.-- Estimated total surface oiled area (1,000s m2) remaining in 

Prince William Sound in 2001. The Stratified random sampling (SRS) mode was used to expand for unexamined beach area and precision was obtained from analytical and bootstrap calculations. The column “Estimate” refers to point estimate if calculation is analytical, or mean estimate, if bootstrap. Lower and upper limits refer to bounds of the 95% confidence interval (CI). 

Computation Method 

Estimate Lower 

95% CI 

Upper 95% CI 

Standard Error 

Coefficient of Variation 

Analytical SRS 42.9 16.1 69.7 13.7 0.32 Bootstrap SRS 41.3 20.7 70.5 13.2 0.32 
41 

Table 6.-- Estimated subsurface oil weight (metric tons) per stratum as determined by three modes of expansion 

for unexamined beach area - Adaptive, Stratified random sampling (SRS), and Observed - and precision from analytical and bootstrap calculations.The column “estimate,” refers to point estimate if calculation is analytical, or mean estimate, if bootstrap. Lower and upper limits refer to bounds of the 95% confidence interval (CI). 

Stratum (h) Computation Method Estimate Lower 

95% CI 

Upper 95% CI 

Standard Error 

Coefficient of Variation 

1. 1990-93, heavy, <100 m Analytical Adaptive 7.7 1.9 13.5 2.95 0.38 

Bootstrap Adaptive 7.6 2.7 13.8 2.88 0.38 Bootstrap SRS 7.5 3 13.8 2.71 0.36 Bootstrap Observed 5.9 1.6 12.2 2.71 0.46 

2. 1990-93, heavy, 100 m Analytical Adaptive 18.3 4.1 32.5 7.25 0.4 

Bootstrap Adaptive 18 6.3 34.7 7.49 0.42 Bootstrap SRS 18.7 6.7 36.6 7.59 0.41 Bootstrap Observed 14.5 3.6 31 7 0.48 

3. 1990-93, med, < 100 m Analytical Adaptive 4 -1.7 9.6 2.88 0.73 

Bootstrap Adaptive 4 0.3 10.9 2.92 0.74 Bootstrap SRS 4.9 0.3 12.4 3.27 0.66 Bootstrap Observed 1.3 0.02 3.3 0.88 0.7 

4. 1990-93, med, 100 m Analytical Adaptive 16 -6.6 38.6 11.5 0.72 
42

Bootstrap Adaptive 16.5 0.5 42.9 11.4 0.69 Bootstrap SRS 17.5 0.5 48.8 13.6 0.78 Bootstrap Observed 11.5 0.01 37 11 0.96 

5. 1989, heavy, < 100 m Analytical Adaptive 3.5 -3.4 10.4 3.53 1 

Bootstrap Adaptive 3.5 0 11.3 3.3 0.94 Bootstrap SRS 7 0 24.1 7.03 1 Bootstrap Observed 3.5 0 10.7 3.21 0.91 
43 

Table 7.-- Estimated total subsurface oil weight (metric tons) remaining in Prince William 

Sound in 2001 as determined by three modes of expansion for unexamined beach area 

- Adaptive, Stratified random sampling (SRS), and Observed - and precision from 

 analytical and bootstrap calculations. The column “estimate,” refers to point estimate 

if calculation is analytical, or mean estimate, if bootstrap. Lower and upper limits 

refer to bounds of the 95% confidence interval (CI). 

Computation Method Estimate Lower 

95% CI 

Upper 

95% CI 

Standard 

Error 

Coefficient 

of Variation 

Analytical Adaptive 49.5 22.4 76.7 13.8 0.28 

Bootstrap Adaptive 49.6 24.4 82.6 15.2 0.31 

Bootstrap SRS 55.6 26.1 94.4 17.9 0.32 

Bootstrap Observed 36.7 14.5 66.4 14 0.38 
44

Figure 1.-- Prince William Sound with Bligh Reef (*), the grounding site of the Exxon Valdez, 

and the beaches affected by the spill. Symbols indicate stations where oil was 

detected (triangles) or not (circles). 
45

Figure 2. -- The total shoreline within any oiling category consisted of discontinuous beach 

segments of varying lengths scattered throughout the spill area. These variable 

length segments were then divided into subsegments of 100 m length or less. In the 

case above, a 183 m segment was divided into a 100 m and a 83 m subsegment. 
46

Upper 

Intertidal A B C D E F G H 

+4.8m 0.5 

1.0 

1.5 

2.0 

2.5 

3.0 

+1.8m 

Lower 

Oil 

12.5m 

Intertidal 100 m 

Figure 3. -- Sampling scheme for a 100 m beach subsegment. The 100 m length beach is 

divided into 8 columns, 12.5 ms wide which are then surveyed for 3 vertical meter 

drops in 0.5 m intervals. Two points are chosen at random within each of the 

resulting 48 blocks and a 0.5 by 0.5 m area is dug down 0.5 m depth and closely 

observed for oil. Therefore 96 pits are excavated in a 100 m beach, barring cliffs or 

bedrock. Fewer columns are sampled if the beach is less than 100 m in length . 

Within each subsegment, column widths are always kept the same width. We used 

the following calculation to determine the number of columns for beaches <100 m 

in length: # of columns = integer (length of beach/12) with a minimum of 1 column 

(if length is less than 12 m). The column width was calculated by dividing the 

beach length by the number of columns determined above and rounding to the 

nearest 0.5 m. 
1

6

2

7

3

8

4 5

9 10 11

Origin

pit

47

0 0

0 0

0 0

0 

0 

0 

0 

0 0 0 

0 

0 

0 

0 

0 0

0 0

0 0

0 1

0

6 

2 

7 

3 

8 

4 5 0

9 10 11 0

0 0 0

0

0

0

0

Origin 

pit 

Figure 4. -- An example of the adaptive sampling methodology. Each square in the grid 

represents a 0.5 by 0.5 m area of a potential pit. The origin pit’s location is chosen 

at random within each block (2 pits per block). If oil is observed in the origin pit, 

four quadrats are excavated to a 0.5 m depth, directly above, below, to the right, and 

the left of the origin pit. If oil is discovered in any of the adaptive pits, more pits are 

dug in the same fashion. Eventually the total area of the oil patch is delineated 

unless the patch extends outside the chosen beach boundaries. In the figure above, 

the numbered squares represent dug pits with 0s being adaptive pits with no oil 

found. 
48

Subsegment estimates: SRS vs Adaptive 

0 

1,000 

2,000 

3,000 

4,000 

5,000 

6,000 

0 1,000 2,000 3,000 4,000 5,000 6,000 

Adaptive 

SRS 

Subsegment estimates: SRS vs Adaptive

0

1,000

2,000

3,000

4,000

5,000

6,000

0 1,000 2,000 3,000 4,000 5,000 6,000

Adaptive

SRS

Figure 5.-- Estimated subsurface oiled areas (quadrats) in sampled subsegments by Stratified 

andom sampling (SRS) and Adaptive modes, and the line of equality between 

modes. 
Subsegment estimates: SRS vs Visual

0

1,000

2,000

3,000

4,000

5,000

6,000

0 1,000 2,000 3,000 4,000 5,000 6,000

Visual

SRS

49

Subsegment estimates: SRS vs Visual 

0 

1,000 

2,000 

3,000 

4,000 

5,000 

6,000 

0 1,000 2,000 3,000 4,000 5,000 6,000 

Visual 

SRS 

Figure 6.-- Estimated subsurface oiled areas (quadrats) in sampled subsegments by Stratified 

random sampling (SRS) compared to the numbers of sampled quadrats with oil 

present (Visual). Line of equality separates known underestimates (below line) from 

feasible estimates (above line). 
50

Subsegment estimates: Adaptive vs Visual 

0 

1,000 

2,000 

3,000 

4,000 

5,000 

6,000 

0 1,000 2,000 3,000 4,000 5,000 6,000 

Visual 

Ad

a

p

tive 

Subsegment estimates: Adaptive vs Visual

0

1,000

2,000

3,000

4,000

5,000

6,000

0 1,000 2,000 3,000 4,000 5,000 6,000

Visual

Ad

a

p

tive

Figure 7.-- Estimated subsurface oiled areas (quadrats) in sampled subsegments by Adaptive 

sampling compared to the numbers of sampled quadrats with oil present (Visual), 

and line of equality. 
Subsegment standard errors: SRS vs Adaptive

0

200

400

600

800

1,000

0 200 400 600 800 1,000

Adaptive

SRS

51

Subsegment standard errors: SRS vs Adaptive 

0 

200 

400 

600 

800 

1,000 

0 200 400 600 800 1,000 

Adaptive 

SRS 

Figure 8.-- Estimated standard errors (quadrats) of subsurface oiled area estimates for sampled 

subsegments, by Stratified random sampling (SRS) and Adaptive modes, and line of 

equality. 

53

Appendices 

55 

Appendix I 

⎛ ⎞ n1 

∑ y$1i ⎜

⎜ ⎟⎟ The estimation equation is T$

1 = N1 

i=1 

, and its variance is given by the general 

n ⎜ 1

⎝ ⎟⎠

formula (see Equation 10 on page 134 of Thompson 1992), 

Var T = [E T s ] + E[var( T s $ ( | )] . $) var ( $| ) 1 1 1 1 1 

. 

N1 

n1 

⎛ ⎞ ⎜

⎝ ⎟

⎠ Here s1 denotes one of the possible selections of n1 100 m beach subsegments 

from the N1 available. The conditional expectation and variance of T$

1 , given a particular 

selection of 100 m beach subsegments, s1, are taken over the possible samples of quadrats 

obtained by adaptive sampling in s1. The unconditional variance and expectation are taken over 

all possible selections of s1. For any 100 m beach subsegment included in sampling (say the ith 

subsegment), the expected value of y$i is yi

 because the adaptive stratified sampling is unbiased. 

Therefore, 

⎛ ⎞ ⎛ ⎞ n1 n1

∑ Eys ( i | 1 ) i=1 ∑ y ⎜ i

⎜

⎜

⎝

⎟

⎟

⎟

⎠

= N1 

⎜

⎜

⎜

⎝

⎟

⎟

⎟

⎠

E T [( 1 s 1 ] $| ) = N = N y 11 . i=1 

1 

n1 n1 

Because of simple random sampling of the 100 m subsegments, the corresponding unconditional 

variance is 
56 

2 y

( $ | )] = var N y s ] = N var( y ) = ( 

σ2 

var [E T s [ | N N − n ) 11 1 1 1 1 1 1 1 1 n1 

where Fy

2

 is the population variance of the unobservable yi, i=1,...,N1. 

The conditional variance of T$

1 , given the selection of 100 m beach subsegments s1, 

depends on the estimation variances for the oiled areas of the individual beach subsegments. Let 

the estimation variance for the ith 100 m beach subsegment be denoted as 

2 ( $ 2 σ y$i = E yi − yi ). . 

Because the estimation errors are independent among beach subsegments, 

⎟

⎟

i 

i=1 1 

∑

⎛ ⎞ n1 

⎜

⎜

⎜

⎜

⎝

y$

n1 

2

⎛ N ⎞ n1 

var( $ |) N ∑ σ 2 T s 11 1 

2 ⎜

⎝ ⎟⎠ = var = ⎟

⎟

⎠

y$i n1 i =1

The unconditional expected value of this conditional variance over all possible selections 

of 100 m beach subsegments is obtained by setting zi

 to be an indicator variable for presence of 

the ith beach subsegment in the random sample (zi = 1 if present, and zi = 0 otherwise) and 

n1 rewriting the preceding conditional variance (noting that the expected value of zi = ) as N1 

2 N N1 ⎛ N ⎞ ⎛ N ⎞ ∑ ∑ 1 1 

n1 

E[var( T s $ | )] 11 = 2 2 E z( i ⎜ )σ σ ⎝ ⎟

⎠ ⎜

⎝ ⎟

⎠ = . y$ y$ i i

i =1 n1 i =1
57 

Appendix II 

Notice that the variance, 

M 

( $ V W ) 2 = ∑ V ( Aj wj ) 2∑ ∑ Cov ( Ajw Akwk + , ). j 

j=1 j k j <

We assume reasonably that Aj and wj  are independent, so that (Goodman 1960) 

V(A w ) = A V () + μ V A + ( V 2 w 

2 () V A ) ( w ). j j j j wj j j j 

Also, for different visual categories, the average weights, wj and wk , are independent and so it 

can be shown by using the definition for covariance that 

Cov ( A w , A w ) = μ μ Cov ( A , A ) . j j k k wj wk j k 

The variance of the visual category means, wj , is V() w = σ2

/ n . Therefore, the variance j wj wj 

of W$

2 is obtained by summing terms, 

)

σ

n 

2 

wi 

wi 

⎫

⎬

⎭

+ 2

M ⎧

⎨

⎩

2 σ wi 

M 

V W( $

2 ) = ∑ ∑ ∑ 2 + μ2 Ai wi V A μ μwi wj ( i V Ai ) ( Cov A ( i Aj + , ). i =1 nwi i =1 j i <

59 

Appendix III 

⎛ p$ k 1 ⎞ ⎜

⎜

⎜

⎝

⎟

⎟

⎟

⎠

The estimate of visual areas in the kth subsegment is a$ y$ M = y$ $ k ⋅pk , where y$ k = ⋅ k k 

p$ kM 

is the estimated oiled area in beach subsegment k (a scalar) and p$ k is the estimated visual area 

composition of subsegment k (a vector). The two terms of a$ k are reasonably considered 

independent. The jth diagonal element of the variance-covariance matrix of estimation errors for 

the kth beach subsegment is defined as 

= [( $ $ ) ] − [E y p ]

2 

v $ E y p 

2 ( $ $ ) . ak −ak , ,jj k kj k kj 

A new expression is obtained by adding and subtracting the term, [E y( $ )] ( $ ) , and k 

2 E p kj 

2 

simplifying the result under the assumption that y$k and p$ kj are unbiased as well as independent, 

v = ( $ )( $ ) − ( $ ) E p + ( $ ) E p − ( $ ) E p a$ −a , ,j j E yk 

2 E p kj 2 [E yk ]

2

( $ kj 

2 ) [E yk ]

2

( $ kj 

2 ) [E yk ]

2 [ ( $ kj )]

2 

k k

= Ep σ + [ ( $ )]

2 p (1 

h 

−

k

p ) 

= y + σ ) p (1 

h 

−

k

p ) 2 2 kj kj 2 2 kj kj 2 2 ( $ ) E y ( ⋅ + σ p . kj y$k k k y$k y$k kj 

The off-diagonal element at the ith row and jth column is obtained by similar expansion of the 

definition, 
60 

v E y p yp ( E y E p p $ = ( $ $ $ $ ) − E yp $ $ ) ( E yp $ $ ) = ( $ 2) ( $ $ ) a −a i j k ki k kj k ki k kj k ki kj k k ,, 

− E( y$ E p $ ) ( Ep $ ) + E( y$ ) ( E p $ ) ( Ep $ ) − ( Ep E p $ ) k 

2

) ( ki kj k 

2 

ki kj [E y $ k )]

2

( $ ki ) ( kj 

p p 2 2 ki kj 2 = − ( y + σ ) hk 

+ σ p p . k y$k y$k ki kj 
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